Using the analogy between the universe and a choral poem, one may view: mathematics as the "rhyme" of the universe. In that perspective new
light is thrown on the unique subject matter of mathematics, the a priori character of its
truth, and the
relation of mathematics to other areas of knowledge. A route is thereby opened for richer use of creativity in mathematics.

Mathematics is the rhyme of the universe. Such is the role I
would assign to mathematics in our understanding of God's
world. This claim makes sense only within a certain framework for understanding the nature of
science.
In two previous
articles (Poythress 1983a 1983b) I have argued that it is
fruitful to consider that the universe is God's poem. Science
exploits some special kind of analogy within the "poem."
Now that same framework will serve also as the larger
framework for my reflections on mathematics.

Within that framework, I declare that mathematics is the
rhyme of the poem. What, then, do I want to suggest by this
analogy? Several things. (1) Mathematics has to do with a
particular subpart or aspect of the total "poem" of the
universe. It cuts across and intersects many other analogies
and metaphors within the poem. (2) Mathematics as rhyme is,
in a sense,
the
most "primitive" analogy in the poem; it is
based on the simple idea of identity and difference. (3) The
possibilities of mathematics-rhyme are deeply bound up with
the nature of the "language-system" as a whole. The properties are given largely a priori by the system, unlike other
analogies in the poem that are included in the poem at the
discretion of the creator. (4) Mathematics as rhyme functions
in the service of
the poem as a whole. It enhances the main
points of the poem, but it is not ultimately intelligible simply
in itself. It is far from having a totally independent purpose.

Let me now consider these points in greater detail.

Mathematics as an Aspect of the "Poem"

First of all, then, mathematics has to do with a particular
subpart or aspect of the total "poem" that is the universe. I
can therefore apply to mathematics some of my general
statements about the universe given in the earlier articles.

Mathematics is (a)
personally
structured, (b) linguisically
structured, (c) shot through with metaphor and
analogy, (d)
utterly dependent on God, (e) characterized by
development
(f) surprising in its victory over chaos.

Almost everything that I said earlier about science can be
applied and worked out in the area of mathematics. I am not
going to proceed straightforwardly to do this working out. But doing so would not be trivial. Philosophies of mathematics have often vigorously denied that mathematics was personal, or dependent on God, or at all characterized by
development. Mathematics, people feel, is somehow unique
among the sciences. Perhaps, they say, it is better not to
classify it as a science at all. Mathematics is "independent of
the world." Perhaps the discoveries in physics, chemistry, and
biology are a "surprising victory over chaos," because we
could imagine it to be otherwise, but mathematics is not
surprising because it could not be otherwise.

Mathematics is indeed different from the sciences. I have
tried to capture some of this intuitive feeling for the "independence" of mathematics by characterizing mathematics as
1.
rhyme." With this characterization I point to the distinction
between mathematics and other sciences. The other sciences
are various kinds of analogies and allegories within the total
poem. Mathematics is rhyme acting in coherence with these
various analogies.

Mathematics as a Distinct Science

I claim that mathematics is a distinct science. It interlocks
with all other sciences, much as rhyme interlocks with and
reinforces the other aspects of a poem. But mathematics is not
reducible to some other science (like psychology), any more
than rhyme is reducible to some other aspect of the poem.

Conversely, other sciences are not reducible to mathematics.
Physics talks about energies, bodies, and motions in the world,
and proposes equations that might have been otherwise. It is
not reducible to mathematics, since the equations of mathematics are true in any "world."

I have already talked about the fallacies and illusions of
such attempted "reductionisms" elsewhere (Poythress 1976a:
48-54). And before me Dooyeweerd (1969) and others in the
cosmonomic school of philosophy engaged in rather extensive
explorations and critiques of reductionisms. It suffices for me
to affirm two complementary truths. First, viewing the
matter more positively, we can say this: reductionisms are
plausible, attractive, even useful and fruitful, because of the
stimulus they give to exploring and exploiting manifold
analogies within the total "poem." Virtually anything,
including mathematics, physics, chemistry, or some subdiscipline within these, can be used as a personal "perspective" for
integrating the whole poem. The truths of the subdiscipline,
by personal choice or preference, serve as a focal point
around which to gather by means of analogy all the rest of the
poem.

Second, we can view the matter negatively. Reductionisms
oversimplify. They wipe out and smash the richness of
meaning in the poem, by a monomania for seeing only one
meaning. Why? The existence of two irreducible aspects of
the poem in harmonious interaction is evidence of a design
and a Designer. Because men would rather flee from God and
hide that evidence f rom themselves, they proclaim that an
impersonal explanation "reducing" one to the other is sufficient explanation.

Mathematics as a priori Truth

My next two points about the nature of mathematics
belong together. First, mathematics is the most "primitive"
type of analogy in the poem. Second, its structure is bound up
with the nature of the "language-system" of the poem. By
these aphorisms or analogies I attempt to indicate both the
unique subject-matter of mathematics and the unique
impression that its truths are a priori.

Let us prepare the ground a little by reflecting on rhyme in
the literal sense. The possibility of rhyme and the characteristics of rhyme in poetry are bound up with structures of
similarity and difference in a language-system. Consider two
words like "love" and "dove." They rhyme if (1) the final
vowel and any subsequent consonants are exactly the same in the two words; (2) the remaining parts of the two words are
not
identical in sound. By this definition "sight" and "site"
are not "rhyming" words but words
identical
in sound
(homonyms). Thus the phenomenon of rhyme derives from
properties of both identity and difference in the phonemic
system or sound system of the language. The potential for
rhyme is "primitive" in the sense that it is based on very
elementary properties of the phonemic system. The phonemic system in turn is the simplest and most basic of the
language systems.

Now let us compare this with mathematics, Mathematics
likewise has to do with properties of identity and difference
in the universe-in God's macropoem. It focuses on the very
most "elementary" properties, the properties of identity and
difference, in the universe. This focus on identity and difference determines its unique perspective or subject-matter.
Simultaneously, that focus helps to explain the apparently a
priori character of mathematical truth. Again let us return to
poetry. The possibilities for different rhyming syllables, for
masculine rhymes, feminine rhymes, imperfect rhymes, and
the like, are given a priori by the language system, before a
poet sets his pen to paper. The monolingual can hardly
conceive of rhyme being other than what it is in his system.
Similarly, in mathematics we are all, in a sense, monolinguals.
We have experience of only one universe. It is difficult to
conceive of an alternative mathematics, because our thoughts
are thoughts
within
a single created "system. " Mathematics is
a statement of the fixed properties of the "rhyming" possibilities of that system.

Am I making all of mathematics a matter of contingent
rather than necessary truth? I appear to be saying that our
inability to imagine things otherwise is a limitation in our
created mind and in the creation around us, but not a
limitation from God's point of view. Is that so? Not necessarily. I am saying that we are finite. Our view of possibility must
not legislate what might be possible for God under vastly
different conditions. But God always acts consistently with his
own nature. It is not true that God can do anything at all. He
cannot lie, he cannot deny himself, he cannot change, and so
on. God does whatever he
wishes
(Ps. 115:3). His wishes are
always consistent with who he is.

Vern Sheridan Poythress is presently Associate Professor of New Testament at
Westminster Theological Seminary. He has a particular interest in interpretive
principles, based on his background in linguistics and apologetics. He holds six
earned degrees, including a Ph.D. in mathematics from Harvard University, a Th.D. in New Testament from the University of Stellenbosch (South Africa), and
masters degrees in biblical studies from the University of Cambridge and
Westminster Theological Seminary. He has also taught linguistics at the University of Oklahoma. He has published a book on Christian philosophy of science,
and articles in the areas of mathematics, philosophy of science, linguistics,
hermeneutics, and biblical studies. Dr. Poythress is a minister in the Presbyterian
Church in America.
Now, this has implications for mathematics. Mathematical
regularities are a reflection of the faithfulness of God. Thus it
may be that a large portion at least of ordinary low-level
mathematics would
necessarily
hold in
any
universe that God
might create. Let me again use the analogy of language.
There is indeed more than one possible human language;
there is more than one language system. But
all
human
language systems have some structural properties in common.
Within certain bounds,
all
are capable of rhyme. All human
languages are characterized by certain constraints because of
the nature of humanity. Analogously, we might say that all
11
systems of possibility" within which God speaks (creates) a
universe-poem are constrained by the nature of God.

In addition to this, there is also at least some degree of
a
posteriori
character in our knowledge of mathematics (cf.
Poythress 1976a: 168-172, 1974: 134-138). A poet's particular
selection of rhymes is still open to him, within the limits of a
particular language. Likewise, even given a "system," God's
choice of
what
particular things will be identical and
different, in
what
particular ways, is open to him.

The Subject Matter of Mathematics

I have already given a preliminary indication of the subject-matter of mathematics by saying that it has to do with
the properties of identity and difference in God's poem. But
this is an oversimplification. More is involved in mathematics
than simply properties of identity and difference. How are
we to set the boundaries to what is mathematics? What is the
difference between mathematics and logic? Between mathematics and mathematical physics? Are statistics and game
theory properly parts of mathematics? How are we to answer
such questions?

It seems to me that such questions about boundaries
partly-but only partly-boil down to "semantic" questions.
There is
more
than one way of drawing a boundary. The
analogy between mathematics and rhyme may once again
illustrate. Rhyme in the narrowest sense is closely related to a
number of other regularities of pattern in poetry. One thinks
of imperfect rhymes (e.g., between "pure" and "fewer"),
assonance and alliteration, poetic meters, extended patterns
of rhyme (e.g., the sonnet), onomatopoeia, homonymy. We
are confronted here with a number of phenomena that can
either be included under a single large umbrella term, or
carefully distinguished from one another. The fineness of the
distinctions depends on the perspective and taste of the
observer. Likewise, "mathematics" may be considered as a
larger or smaller area of investigation.

"Mathematics" as a term may be used to cover a larger or
smaller area. I think that I come somewhere near the
ordinary scope of the word "mathematics" when I say that
mathematics has to do with three or four interlocking areas of
investigation, together with the relations between these areas
and their ramifications. These areas are (a) properties of
identity and difference, (b) properties of quantities, (c)
properties of space, and (d) properties of motion. The study of
these areas leads to corresponding academic disciplines: (a)
elementary set theory, concerning the properties of aggregates ("agorology"), (b) number theory and elementary
alge
bra, (c) geometry, and (d) kinematics (see Poythress 1976a:
179-180). Kinematics is usually not considered to be part of
mathematics, but I judge that the limit concept in calculus
depends ultimately on intuitions about motion. Hence it
seems to me that there is much in the field of mathematical
analysis that interacts directly with a somewhat redefined
conception of kinematics.

At any rate, all agree that mathematics has now advanced
to an impressive depth and complexity, partly by studying
higher-level regularities involved in agorology, number theory, and geometry. partly by studying the regularities in the
interactions and 'interconnections between the three or four
fields. It is not in,,- purpose, then, to off er a detailed or
definitive classification of higher reaches of mathematics. My
intent is to suggest some of the sources for mathematics.
Mathematics finds its. sources in various types of intuition
about primitive properties of the universe: identity, quantity,
space, motion.1

This bare-bones account of the nature of the subject-matter
of mathematics needs to be filled out in two directions: the
relation of mathematics to kindred disciplines such as logic,
linguistics, and psychology: and more about further subdivisions within mathematics. and the possibilities of "reducing"
one subdivision to another.

How is mathematics re6ted to logic, to linguistics, and to
psychology? Some philcisophers of mathematics have gone so
far as to claim that mathematics is actually a subdivision of
logic (logicism), or of linguistics ~ formalism), or of psychology
(intuitionism). I, on the contrary, have argued above that
mathematics has a subject-matter of its own distinct from any
of these fields. If I am to justify that claim more thoroughly, I
should give some account of the plausibility of these competing claims.

Global Basis for Plausibility of Reductionisms

To give such an account in general terms is not too
difficult. Mathematics forms one aspect of the universeas-poem. From this I have already inferred that mathematics
is personally structured and linguistically structured. Since
mathematics is linguistically structured, it should be no
surprise that formalism in the philosophy of mathematics has
tried to reduce mathematics to language pure and simple:
"
mathematics is the study of formal languages."

Likewise, mathematics is personally structured. For intelligibility, there must be a personal interpreter. Hence, it is not
surprising that intuitionism in the philosophy of mathematics
has tried to reduce mathematics to a branch of psychology: "
mathematics is the study of
mental
mathematical constructions. "

To explain the basis of logicism is not quite so easy. We
could start with the motif of God as a
person
who is
self-consistent in all that he does, or with the motif of
language as a self-consistent organized system, or with the
motif of victory over chaos. God's poem is not a chaos.
Because of this, we can make inferences and predictions from
observations about some aspects of the poem, and have them
vindicated by other aspects. Order, regularity, and the possibility of inference pervade the poem. Hence mathematics as
a particular aspect of the poem is subject to inference. In fact,
mathematics as the study of very "primitive" properties of
the "poem" is easier to subject to detailed inferential patterns
than are academic disciplines whose subject-matter is less
primitive. Hence the plausibility of saying, "Mathematics is a
branch of logic."

So far I have said nothing about the fourth of the "classical"
positions in philosophy of mathematics: empiricism. Empiricism says that mathematics is a generalization from experience of the physical world. My root metaphor of mathematics
as rhyme accounts for this almost automatically. As rhyme

Let us consider mathematical truth not as simply unproblematically "there,"
but as a victory over chaos, in fact a constantly reasserted victory.

occurs in a poem, so mathematics "occurs" or rather "holds
true in particular cases" in the world. Is my own position,
then, simply a variation on empiricism? Almost, but not
quite. Remember that I argued that mathematics, at least
from a human point of view, has largely an a priori character
because of its interest in the "language-system" behind any
possible piece of the poem. The old empiricism did not
account for this
a
priori element. Nor did it account for the
compatibility between the intuitions of the human mind and
the empirical facts "out there."

The Usefulness and "Success" of Reductionisms

The attractiveness of reductionisms can be understood
even better using the idea of multiple perspectives developed
in my earlier article (Poythress 1983b). According to this idea,
the same subject-matter can frequently be explained or
systematized using more than one point of view. More than
one root metaphor, more than one "model," can sometimes
be developed. In the course of development, there is a kind of
reciprocal interaction between the principal subject (the
thing modeled) and the subsidiary subject (the model used).
The structure of the subsidiary subject stimulates the investigator to try to extend and deepen the model in certain
directions. Contrariwise, the structure of the principal subject
causes modifications, tinkerings, closer definitions, and
ad
hoc
additions to the model. The modifications of the model
enable it to survive when unpalatable evidence shows up.
(For a detailed account of this process, see Kuhn (1970),
Lakatos (1978).)

The four classical philosophies of mathematics can themselves be considered as instances of this type of development.
For all four of them, the works of mathematicians are the
subject-matter, the principal subject. But the four use different root-metaphors (cf. Pepper 1970) or subsidiary subjects
as models for making intelligible this principal subject. For logicism, the subsidiary subject is logic. For formalism, it is
language. For intuitionism, it is the human mind and its
psychology. For empiricism, it is certain physical aspects of
the nonhuman world. The "success" of the four philosophies
simply demonstrates the fruitfulness of considering mathematics from each of the four viewpoints or perspectives. It
demonstrates, in other words, the fruitfulness of a certain
analogy or correspondence.

In the process, there is a mutual enrichment. On the one
hand, the principal subject, mathematics, is better understood
as people try to reexpress it in logical terms, in formalist
terms, etc. On the other hand, there is also modification of the
subsidiary subject. Logic, language, psychology, and physics
are each "enlarged" beyond their former boundaries in the
attempt to encompass mathematics. For instance, logicism
and formalism must each include specifically
mathematical
axioms in their foundations (such as the axiom of infinity and
the axiom of reducibility in the Whitehead-Russell system).
And the reader must know how to interpret or apply.. certain
theorems in a mathematical sense, if he is to profit from
them.

Intuitionism and empiricism have difficulties of a somewhat different kind. In common forms of intuitionism and
empiricism, a great deal of classical mathematics must be
abandoned or modified because it is nonintuitive or nonempirical. Alternatively, the concepts of mathematical "intuition" and of the "empirical" can be boldly and imaginatively
expanded to encompass the full range of what mathematicians do. But then, after this radical expansion, is anything
worthwhile left of the original attempt at reduction?

The Failure of Reductionisms to Deal
with Multiple Perspectives

Reductionist philosophies of mathematics, then, are stimulating as metaphors, but inadequate as ultimate explanations.
I do not intend to review here the criticisms of reductionist
philosophies already put forth by rival reductionisms (cf.
Benacerraf-Putnarn
1964)
or by antireductionist philosophies (Dooyeweerd
1969,
Vollenhoven
1918, 1936,
Strauss
1970,
1971, 1973, Poythress 1974, 1976a, 1976b). Beyond these
criticisms, two more points need to be made.

First, philosophy of mathematics needs to account not only
for mathematics but for the plurality of plausible philosophies
of mathematics! I would argue that nothing short of a
multiperspective approach to mathematics will succeed here.
As in linguistics (cf. Pike 1967:68-72, 84-92, 1980), so in
mathematics, a multiperspective approach is needed to do
justice to both the subjective and objective poles at work in the
subject-area. On the subjective side, the subject's choice of a
perspective, a root-metaphor, or a paradigm as a startingpoint for systematizing his understanding is decisive for the
final form his theory will take. On the objective side, the fact
that the universe as God's poem includes many built-in
metaphors forms the basis for successful development of more
than
one
explanatory model.

Second, using certain insights from Paul Benacerraf (1965),
we can show simultaneously the fruitfulness of multiperspective thinking and the failure of reductionisms. I have in mind
an article by Benacerraf entitled "What Numbers Could Not
Be." What is the point of Benacerraf's article? In brief,
Benacerraf argues that we know for certain that numbers are
not
sets. Rather, there exists, on the basis of a set theory, the
possibility of establishing a stipulatory correlation between
numbers and certain infinite recursive progressions of sets.
There is a correlation (an analogy) rather than a metaphysical
identity. The fact that there is more than one way to establish
a correlation shows that it is a correlation and not an
identity.

Benacerraf's argument is thus an antireductionist argument ("numbers are not sets") based on the use of multiperspectives (multiple possible correlations between sets and
numbers). Benacerraf also uses multiple perspectives more
positively. By examining which correlations between numbers and sets "do the job," he helps to determine what is
"
essential" to number. Many different set-theoretic definitions effectively "capture" the usual relevant properties of
the natural numbers.2 What numbers "are" is what is common to all these capturing correspondences.

Thus a capture of this kind, impressive though it may be,
is still not a metaphysical identity. it is not a reduction in
every sense, since a real
total
reduction would leave us with a
triviality, a tautology: A = A. And of course we may
sometimes find that we did not capture everything we
thought we did. Counterintuitive results in axiomatic set
theory or analysis show us that we didn't capture everything
in our intuition.

Now let us apply a similar technique to the four reductionist philosophies of mathematics. Logicism can be seen to be
inadequate, because there is more than one way of embedding mathematics in logic. Numbers can be represented in
more than one way by sets, as we have seen. And sets can be
represented in more than one way in logical formalism.

Similarly, formalism fails for much the same reason. If
formalism tries to include a theory of the relations between
formal theories, it can do so only by a regress of metalanguages.

Intuitionism is not so easily criticized in this fashion. The
genius of intuitionism is, in fact, to insist that numbers (and
perhaps space) are
sui generis.
But multiple perspectives still
challenge intuitionism more indirectly. Can intuitionism
account for the
existence
of multiple correlations between
(say) the number-system-as-intuited and recursive sequences
of sets obeying the Peano axioms? Can it deal with the
multiplicity of different people's senses of mathematical
"intuition," ranging from extreme finitists to formalists who
temporarily adopt formalized intuitionist logic?

Empiricism is also subject to criticism using multiple
correlations. Straightforward empiricism in mathematics
establishes a correlation between numbers and collections of
objects. "Four" is a kind of generalization from experiences of
collections of four apples, four fingers, etc. But one can
establish correlations in a different way. "Four can be applied
to collections of abstractions ("second-order collections") as
well as collections of "things" ("first-order collections").
(2, 5, 7, 81 is a collection of four numbers; red, green, blue, brown is a collection of four colors. Can empiricism account
for such a generalization cutting across "types"? There is
another problem. "Four" can apply to collections that can be
divided in more than one way. Four pairs of shoes are also
eight shoes; four limbs are one body. Numbers are not
-given " in the world in any simple way. They require the
subjective contribution of a personal interpreter making
decisions as to what differences and identities are relevant to
his interests. Four pairs of shoes can be either an instance of
four or an instance of eight, depending on the perspective.

Multiple Correlations in the Subparts
ofMathematics

Using Benacerraf's principle of multiple correlations, we
can also construct arguments for showing the Donreducibility
of various subparts of mathematics to one another. To provide
a first set of examples, let us focus on the four subareas of
mathematics already distinguished. Mathematics deals with
(a) identity and difference, (b) quantity, (c) space, and (d)
motion. Can we show that these four are not reducible to one
another?

Benacerraf's original argument already shows that numbers cannot be equated with sets. Hence (b) is not reducible to
(a). Second, space is not reducible to set theory, since more
than one set-theoretic formulation can represent the same
geometry. Space is not reducible to number, since there is
more than One way of coordinatizing a space. What about the
reduction of motion to space or quantity? The same motion
can be represented quantitatively in more than one way,
depending on the choice of time coordinate. We need to
choose both the point of origin for the coordinate and the
scale of measurement. Moreover, in order to represent motion
in purely spatial terms, quantitative time must be transformed into another spatial dimension. Again this can be
done in more than one way.

This pattern of argument is in fact capable of demonstrating still further irreducibilities. Ordered pairs are not reducible to sets, since more than one stipulative definition will
work. Nor are functions reducible to sets of ordered pairs.
Groups are not reducible to an ordered triple consisting of a
set, a binary operation of multiplication, and a unary operation of inverse. For groups can also be defined starting from a
set with a single binary operation of multiplication (inverse
being defined only later in the group axioms). Or groups can
be defined using the single binary operation f (a, b) =_ a - b

instead of the binary operation g (a, b) =- a - b.

Radical Irreducibility

If one approaches every area of mathematics in this way,
one is well on the way to a radical extension of the idea of
irreducibility. Up to now, I have applied the idea of irreducibility only to broad areas of study. Quantity, space, and
motion represent such broad areas. But irreducibility can also
be used in narrower cases. From my point of view, nothing is
"identical to" or "reducible to" anything else. To take a most
outrageous example: the number
12
is not "reducible to"
11 + 1. (It could be defined not only as 11 + I but as
10 + 2,
9 + 3, 2 x 6,
etc. Hence none of these is the correct definition
from the point of view of logical deduction.)

To be sure, in many cases stipulative definitions are
capable of serving as a starting point for deducing all the
important properties of the entity so defined (the definiendum), For example, the stipulative definition
12 = 11 + I
can be the starting point, in the context of the Peano axioms,
for deducing the properties of
12.
But that only shows that
there is a detailed analogy, not an identity, between definiendum (e.g.,
12)
and definiens (structures used to do the
defining, e.g., 11 + 1 and Peano axioms). Moreover, it should
be noted that a definition like
12 = 13 + 1
will work only in
the context of a surrounding mathematical system-an axiom
system or its informal equivalent. Not every such definition
would work in every context. Hence we may say that the
definiens and the definiendum are serving respectively as
the subsidiary subject and the principal subject of a mathematical "allegory." Stipulatory definitions are the starting
points for so many allegories. The surrounding mathematical
system furnishes the contextual control for understanding any
particular piece of the allegory.

I do not say that this is the only way of looking at
mathematical definition. But it is useful for several purposes.
I now focus on two of these purposes.

Awakening Wonder
First, I intend by this "allegorical" approach to reawaken
our awareness of wonder in mathematics. We know that it is
useful to consider functions as ordered pairs, or to coordina
tize Euclidean space. This is something to be wondered at.
Even the deducibility of properties of
12
from
12 = 11 + 1 is
ultimately mysterious (cf. Wittgenstein
1967:13-16).
It is
something to praise God for. It is not simply a bare identity
calling for no reaction, or "So what?" Our response can be
wonder, whether or not the truths in question are a priori or a
posteriori
from one or another point of view. For in either
case they are rooted in the wisdom of God.

Consider by contrast the effect of the pronouncement that
"
of course it works." The person says, "Of course," because
"functions are nothing but ordered pairs in the first place," or
coordinatizability is merely the inevitable consequence of
Euclidean axioms," or 12 is
nothing but an alternate name
for 11 + 1. Even if these statements were truer than they
are, they would be an evasion of the ultimately personal
character of creation originating in a creator. To repeat what
I have said before: let us consider mathematical truth not as
simply unproblematically "there," but as a victory over
chaos, in fact a constantly reasserted victory.

Awakening Creativity

My second purpose in using an "allegorical" approach is to
stir creativity. Once the spell of "ordinariness" is broken, we
can let our imaginations play and find alternate "allegories."
When we allow ourselves to imagine what it would be like for
the original allegory to break down, we are freed to produce
creative alternatives. We may find, for example, non-Euclidean geometries, fuzzy functions (cf. Zadeh 1956, WangChang 1980), or alternate number systems.

Independent of my own thinking on creativity, William J.
Gordon (1961) has developed a theory of creativity emphasizing personal involvement, empathy, fantasy, and emotions as
useful aids in technological invention and business. He is
much more specific about techniques of creativity than I can
be here. But we have both emphasized the involvement of the
person
of the investigator in knowledge. I can illustrate bow
this works by taking as an example the positive integers. How
can there be creativity here, since the facts are (apparently) so
cut-and-dried? Well, there is of course creativity involved in
the discovery of new proofs in number theory. But I want to
exercise creativity on a far more basic level.

To do so I personify the integers. I visualize not an infinite
series of bare symbols 1, 2, 3, 4, . . . , but a row of people. The
successor relation I visualize by having each person lay his
hand on the shoulder of the next one, or by having each
person throw a ball to the next one. Then I fantasize about the
ways in which the number system could break down or
behave differently. What could happen? All sorts of things.
The people could form themselves into a circle instead of a
straight line. We would have modular arithmetic. Or at
certain points the line could split in two, and we would have a
discrete partial ordering. I could imagine each person juggling many balls instead of just one which he passes to the
next. Then we have the beginning of the concept of order
pairs. I could imagine running out of persons to continue the
line, so that the last person had to keep his ball. This
corresponds to the finite universe that Whitehead and Russell
had to eliminate with their axiom of infinity.

Mathematical Meaning as Meaning in Relationship

Finally, my "allegorical" approach or "poetic" approach to
mathematics also encourages a useful emphasis on the
relational
aspect of mathematical truth and mathematical understanding. What do I mean by relational aspect? To understand and appreciate a truth of mathematics is to understand
it
in relation to
many other truths both inside and outside the
area of mathematics. (Cf. earlier claims to this effect in
Poythress 1976b:172-173.)

In poetry, rhyme finds its significance, its effectiveness, its
raison d'etre,
not purely in itself but in its functions in the
larger whole. Likewise mathematical truth finds its significance not merely in itself, but in relation to applications and
parallels in other areas of mathematics, plus applications in
physics, economics, and still other areas. Of course, I want to
affirm vigorously that the attempt to "purify" mathematics,
to isolate general principles from the specific practical contexts in which they first appeared, has been quite fruitful. But
the preference for pure abstraction over concrete embodiment is both one-sided and ineffectual, from a pedagogical as
well as a philosophical point of view. Teachers know very
well that group theory is best learned when worked-out
examples of particular groups are sprinkled in with theorems.
Calculus is best learned when examples with particular
functions accompany its theorems.

Moreover, the best tests of mathematical knowledge come
through applications. For instance, a student who can quote
the theorems, explain their meaning, and even repeat the
proofs still does not really "know" calculus or group theory
unless he can work problems. I would suggest that it is best to
treat this pedagogical fact as a fact constitutive for the
nature
of mathematical truth. It is not simply an inconvenient
limitation, a falling short of the Platonic ideal, a concession to
the limited powers of men of dust. Remember that Plato was
against the body and its "messy" corruption of the pure vision
of the abstract ideal. Plato was against creation, in fact. But a
Christian ought not to be. The pedagogical constraints are not
.1
unfortunate" corruptions, but an aspect of the created
structure of mathematical knowledge.

Pedagogically, then, I am in favor of the reintroduction of
the writhing dirty masses of applications into mathematical
explanation. One can still keep the abstract generalizations
with their Apollonian beauty. But the particular examples are
not to be "reduced" to the generality. We ought to revive our
wonder for the fact that the generality actually
holds
for this
case, and for that case, and for this other case. Each discovery
of a new application can be seen as a development of
mathematical truth, the writing of a new line to the poem.

NOTES

1If one is willing to apply a good deal of imagination, one can work out the
analogy between mathematics and poetic rhyme even to include this detail.
Properties of identity and difference in mathematics correspond to the
identity and difference necessary for true rhyme. Properties of quantity
correspond to meter in poetry, with its quasiquantitative count of feet.
Properties of space correspond to the structural patterns of regular
rhyming schemes (e.g-, the sonnet).